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arXiv · 1303.1777

Epsilon-complexity of continuous functions

Boris Darkhovsky·Alexandra Pyriatinska
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Abstract

A formal definition of epsilon-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov's idea of the complexity of an object. A definition of epsilon-complexity for a class of continuous functions with a given modulus of continuity is also proposed. Additionally, an explicit formula for the epsilon-complexity of a functional class is obtained. As a consequence, the paper finds that the epsilon-complexity for the Holder class of functions can be characterized by a pair of real numbers. Based on these results the papers formulates a conjecture concerning the epsilon-complexity of an individual function from the Holder class. We also propose a conjecture about characterization of epsilon-complexity of a function from the Holder class given on a discrete grid.

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Boris Darkhovsky, Alexandra Pyriatinska. 2013-02-08. Epsilon-complexity of continuous functions. https://arxiv.org/abs/1303.1777

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